| L(s) = 1 | + (2.78 − 0.491i)2-s + (−2.20 − 2.63i)3-s + (7.51 − 2.73i)4-s + (27.1 + 9.88i)5-s + (−7.44 − 6.24i)6-s + (21.0 − 36.4i)7-s + (19.5 − 11.3i)8-s + (12.0 − 68.1i)9-s + (80.5 + 14.1i)10-s + (17.9 + 31.0i)11-s + (−23.8 − 13.7i)12-s + (−193. + 230. i)13-s + (40.7 − 111. i)14-s + (−33.9 − 93.3i)15-s + (49.0 − 41.1i)16-s + (61.5 + 348. i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (−0.245 − 0.292i)3-s + (0.469 − 0.171i)4-s + (1.08 + 0.395i)5-s + (−0.206 − 0.173i)6-s + (0.429 − 0.744i)7-s + (0.306 − 0.176i)8-s + (0.148 − 0.841i)9-s + (0.805 + 0.141i)10-s + (0.148 + 0.256i)11-s + (−0.165 − 0.0954i)12-s + (−1.14 + 1.36i)13-s + (0.207 − 0.571i)14-s + (−0.150 − 0.414i)15-s + (0.191 − 0.160i)16-s + (0.212 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.19296 - 0.535439i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19296 - 0.535439i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.78 + 0.491i)T \) |
| 19 | \( 1 + (316. + 173. i)T \) |
| good | 3 | \( 1 + (2.20 + 2.63i)T + (-14.0 + 79.7i)T^{2} \) |
| 5 | \( 1 + (-27.1 - 9.88i)T + (478. + 401. i)T^{2} \) |
| 7 | \( 1 + (-21.0 + 36.4i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-17.9 - 31.0i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (193. - 230. i)T + (-4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-61.5 - 348. i)T + (-7.84e4 + 2.85e4i)T^{2} \) |
| 23 | \( 1 + (-39.9 + 14.5i)T + (2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (989. + 174. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-483. - 278. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 598. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.10e3 - 1.31e3i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-749. - 272. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-649. + 3.68e3i)T + (-4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + (-463. - 1.27e3i)T + (-6.04e6 + 5.07e6i)T^{2} \) |
| 59 | \( 1 + (4.58e3 - 808. i)T + (1.13e7 - 4.14e6i)T^{2} \) |
| 61 | \( 1 + (1.89e3 - 689. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-5.81e3 - 1.02e3i)T + (1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-1.39e3 + 3.84e3i)T + (-1.94e7 - 1.63e7i)T^{2} \) |
| 73 | \( 1 + (5.26e3 - 4.42e3i)T + (4.93e6 - 2.79e7i)T^{2} \) |
| 79 | \( 1 + (-996. - 1.18e3i)T + (-6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-5.50e3 + 9.54e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-5.00e3 + 5.97e3i)T + (-1.08e7 - 6.17e7i)T^{2} \) |
| 97 | \( 1 + (-1.59e4 + 2.80e3i)T + (8.31e7 - 3.02e7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.95613421664380220462543123791, −14.28918847728802732047438976795, −13.14274945477565037378189914380, −12.00799393462798056863261925078, −10.66198499458289804888020519737, −9.453299454364169327953822861826, −7.13090829555554349099843199382, −6.12344085283909302191235673620, −4.27634418827802748586521384978, −1.90372387092495166640478280439,
2.36138113314328916279969380310, 4.98965476100263205136678020064, 5.73019865599685244949886329843, 7.78993509274263712933813756730, 9.524005621787769885859332140969, 10.79808179511934594039700042417, 12.26396537264278012799352130965, 13.28935071012687585881320517468, 14.41044883390025702904732035468, 15.52561113677626495333748171381
